Skip to main content
Global

4.13: Tatua equations na FRACTIONS (Sehemu ya 2)

  1. Last updated
  2. Save as PDF
  • Page ID
    173382

    • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    Tatua Ulinganisho na Mgawo wa Sehemu

    Tunapokuwa na equation na mgawo wa sehemu tunaweza kutumia Mali ya Kuzidisha ya Usawa ili kufanya mgawo sawa na\(1\). Kwa mfano, katika equation:

    \[\dfrac{3}{4}x = 24 \nonumber \]

    Mgawo wa\(x\) ni\(\dfrac{3}{4}\). Ili kutatua\(x\), tunahitaji mgawo wake kuwa\(1\). Kwa kuwa bidhaa ya idadi na usawa wake ni\(1\), mkakati wetu hapa utakuwa kujitenga\(x\) na kuzidisha kwa usawa wa\(\dfrac{3}{4}\). Tutafanya hivyo katika Mfano\(\PageIndex{1}\).

    Mfano\(\PageIndex{8}\): solve

    Kutatua:\(\dfrac{3}{4}x = 24\).

    Suluhisho

    Panua pande zote mbili kwa usawa wa mgawo. \(\textcolor{red}{\dfrac{4}{3}} \cdot \dfrac{3}{4} x = \textcolor{red}{\dfrac{4}{3}} \cdot 24 \)
    Kurahisisha. \(1x = \dfrac{4}{3} \cdot \dfrac{24}{1} \)
    Kuzidisha. \(x = 32 \)

    Angalia:

    Mbadala x = 32. \(\dfrac{3}{4} \cdot 32 \stackrel{?}{=} 24 \)
    Andika upya 32 kama sehemu. \(\dfrac{3}{4} \cdot \dfrac{32}{1} \stackrel{?}{=} 24 \)
    Kuzidisha. Equation ni kweli. \(24 = 24 \; \checkmark\)

    Kumbuka kwamba katika equation\(\dfrac{3}{4} x = 24\), tunaweza kuwa na kugawanywa pande zote mbili\(x\) na\(\dfrac{3}{4}\) kupata yenyewe. Kugawanyika ni sawa na kuzidisha kwa usawa, kwa hiyo tutapata matokeo sawa. Lakini watu wengi wanakubaliana kwamba kuzidisha kwa usawa ni rahisi.

    Zoezi\(\PageIndex{15}\)

    Kutatua:\(\dfrac{2}{5}n = 14\).

    Jibu

    \(35\)

    Zoezi\(\PageIndex{16}\)

    Kutatua:\(\dfrac{5}{6}y = 15\).

    Jibu

    \(18\)

    Mfano\(\PageIndex{9}\): solve

    Kutatua:\(− \dfrac{3}{8}w = 72\).

    Suluhisho

    Mgawo ni sehemu hasi. Kumbuka kwamba idadi na usawa wake wana ishara sawa, hivyo usawa wa mgawo lazima pia uwe hasi.

    Kuzidisha pande zote mbili kwa usawa wa\(− \dfrac{3}{8}\). \(\textcolor{red}{- \dfrac{8}{3}} \left(- \dfrac{3}{8} w \right) = \left(\textcolor{red}{- \dfrac{8}{3}}\right) 72 \)
    Kurahisisha; kurudia kuzidisha kwa moja. \(1w = - \dfrac{8}{3} \cdot \dfrac{72}{1} \)
    Kuzidisha. \(w = -192\)

    Angalia:

    Hebu w = -192. \(- \dfrac{3}{8} (-192) \stackrel{?}{=} 72 \)
    Kuzidisha. Ni hundi. \(72 = 72 \; \checkmark \)
    Zoezi\(\PageIndex{17}\)

    Kutatua:\(− \dfrac{4}{7}a = 52\).

    Jibu

    \(-91\)

    Zoezi\(\PageIndex{18}\)

    Kutatua:\(− \dfrac{7}{9}w = 84\).

    Jibu

    \(-108\)

    Tafsiri sentensi kwa equations na Kutatua

    Sasa tuna kufunikwa mali zote nne za usawa-kutoa, Aidha, mgawanyiko, na kuzidisha. Tutaweza orodha yao yote pamoja hapa kwa ajili ya kumbukumbu rahisi.

    Jedwali\(\PageIndex{2}\)
    Ondoa Mali ya Usawa: Kwa idadi yoyote halisi a, b, na c, ikiwa = b, basi - c = b - c. Kuongeza Mali ya Usawa: Kwa idadi yoyote halisi a, b, na c, ikiwa = b, basi a + c = b + c.
    Idara ya Mali ya Usawa: Kwa idadi yoyote a, b, na c, ambapo c ∙ 0 ikiwa = b, basi\(\dfrac{a}{c} = \dfrac{b}{c}\). Kuzidisha Mali ya Usawa: Kwa idadi yoyote halisi a, b, na c ikiwa = b, basi ac = bc.

    Unapoongeza, ondoa, kuzidisha au ugawanye kiasi sawa kutoka pande zote mbili za equation, bado una usawa. Katika mifano michache ijayo, tutaweza kutafsiri sentensi katika equations na kisha kutatua equations. Inaweza kuwa na manufaa kupitia meza ya tafsiri katika Tathmini, Kurahisisha, na kutafsiri Maneno.

    Mfano\(\PageIndex{10}\): solve

    Tafsiri na kutatua:\(n\) kugawanywa na\(6\) ni\(−24\).

    Suluhisho

    Tafsiri.
    Panua pande zote mbili kwa 6. \(\textcolor{red}{6} \cdot \dfrac{n}{6} = \textcolor{red}{6} (-24)\)
    Kurahisisha. \(n = -144 \)
    Angalia: Je, -144 imegawanywa na 6 sawa na -24?
    Tafsiri. \(\dfrac{-144}{6} \stackrel{?}{=} -24\)
    Kurahisisha. Ni hundi. \(-24 = -24 \; \checkmark \)
    Zoezi\(\PageIndex{19}\)

    Tafsiri na kutatua:\(n\) kugawanywa na\(7\) ni sawa na\(−21\).

    Jibu

    \(\dfrac{n}{7} = -21\);\(n=-147\)

    Zoezi\(\PageIndex{20}\)

    Tafsiri na kutatua:\(n\) kugawanywa na\(8\) ni sawa na\(−56\).

    Jibu

    \(\dfrac{n}{8} = -56\);\(n=-448\)

    Mfano\(\PageIndex{11}\): solve

    Tafsiri na kutatua: quotient ya\(q\) na\(−5\) ni\(70\).

    Suluhisho

    Tafsiri.
    Panua pande zote mbili kwa -5. \(\textcolor{red}{-5} \left(\dfrac{q}{-5}\right) = \textcolor{red}{-5} (70) \)
    Kurahisisha. \(q = -350\)
    Angalia: Je, quotient ya -350 na -5 ni sawa na 70?
    Tafsiri. \(\dfrac{-350}{-5} \stackrel{?}{=} 70 \)
    Kurahisisha. Ni hundi. \(70 = 70 \; \checkmark \)
    Zoezi\(\PageIndex{21}\)

    Tafsiri na kutatua: quotient ya\(q\) na\(−8\) ni\(72\).

    Jibu

    \(\dfrac{q}{-8} = 72\);\(q=-576\)

    Zoezi\(\PageIndex{22}\)

    Tafsiri na kutatua: quotient ya\(p\) na\(−9\) ni\(81\).

    Jibu

    \(\dfrac{p}{-9} = 81\);\(p=-729\)

    Mfano\(\PageIndex{12}\): solve

    Tafsiri na kutatua: Theluthi mbili ya\(f\) ni\(18\).

    Suluhisho

    Tafsiri.
    Kuzidisha pande zote mbili na\(\dfrac{3}{2}\). \(\textcolor{red}{\dfrac{3}{2}} \cdot \dfrac{2}{3} f = \textcolor{red}{\dfrac{3}{2}} \cdot 18 \)
    Kurahisisha. \(f = 27 \)
    Angalia: Je, theluthi mbili ya 27 sawa na 18?
    Tafsiri. \(\dfrac{2}{3} (27) \stackrel{?}{=} 18\)
    Kurahisisha. Ni hundi. \(18 = 18 \; \checkmark \)
    Zoezi\(\PageIndex{23}\)

    Tafsiri na kutatua: Mbili-tano ya\(f\) ni\(16\).

    Jibu

    \(\dfrac{2}{5}f = 16\);\(f=40\)

    Zoezi\(\PageIndex{24}\)

    Tafsiri na kutatua: Tatu-nne ya\(f\) ni\(21\).

    Jibu

    \(\dfrac{3}{4}f = 21\);\(f=28\)

    Mfano\(\PageIndex{13}\): solve

    Tafsiri na kutatua: quotient ya\(m\) na\(\dfrac{5}{6}\) ni\(\dfrac{3}{4}\).

    Suluhisho

    Tafsiri. \(\dfrac{m}{\dfrac{5}{6}} = \dfrac{3}{4} \)
    Kuzidisha pande zote mbili\(frac{5}{6}\) kwa kutenganisha m. \(\dfrac{5}{6} \left(\dfrac{m}{\dfrac{5}{6}}\right) = \dfrac{5}{6} \left(\dfrac{3}{4}\right) \)
    Kurahisisha. \(m = \dfrac{5 \cdot 3}{6 \cdot 4}\)
    Ondoa mambo ya kawaida na uongeze. \(m = \dfrac{5}{8} \)

    Angalia:

    Je quotient ya\(\dfrac{5}{8}\) na\(\dfrac{5}{6}\) sawa na\(\dfrac{3}{4}\)? \(\dfrac{\dfrac{5}{8}}{\dfrac{5}{6}} \stackrel{?}{=} \dfrac{3}{4} \)
    Andika upya kama mgawanyiko. \(\dfrac{5}{8} \div \dfrac{5}{6} \stackrel{?}{=} \dfrac{3}{4} \)
    Panua sehemu ya kwanza kwa usawa wa pili. \(\dfrac{5}{8} \cdot \dfrac{6}{5} \stackrel{?}{=} \dfrac{3}{4} \)
    Kurahisisha. \(\dfrac{3}{4} = \dfrac{3}{4} \; \checkmark \)

    Ufumbuzi wetu hundi.

    Zoezi\(\PageIndex{25}\)

    Tafsiri na kutatua. Quotient ya\(n\) na\(\dfrac{2}{3}\) ni\(\dfrac{5}{12}\).

    Jibu

    \(\dfrac{n}{\dfrac{2}{3}} = \dfrac{5}{12}\);\(n = \dfrac{5}{18}\)

    Zoezi\(\PageIndex{26}\)

    Tafsiri na kutatua. Quotient ya\(c\) na\(\dfrac{3}{8}\) ni\(\dfrac{4}{9}\).

    Jibu

    \(\dfrac{c}{\dfrac{3}{8}} = \dfrac{4}{9}\);\(c = \dfrac{1}{6}\)

    Mfano\(\PageIndex{14}\): solve

    Tafsiri na kutatua: Jumla ya tatu na nane na\(x\) ni tatu na nusu.

    Suluhisho

    Tafsiri.
    Tumia Mali ya Kuondoa ya Usawa ili\(\dfrac{3}{8}\) uondoe pande zote mbili. \(\dfrac{3}{8} + x - \dfrac{3}{8} = 3 \dfrac{1}{2} - \dfrac{3}{8} \)
    Kuchanganya kama maneno upande wa kushoto. \(x = 3 \dfrac{1}{2} - \dfrac{3}{8} \)
    Badilisha nambari iliyochanganywa kwa sehemu isiyofaa. \(x = 3 \dfrac{1}{2} - \dfrac{3}{8} \)
    Badilisha kwa sehemu sawa na LCD ya 8. \(x = \dfrac{7}{2} - \dfrac{3}{8} \)
    Ondoa. \(x = \dfrac{25}{8} \)
    Andika kama nambari iliyochanganywa. \(x = 3 \dfrac{1}{8} \)

    Tunaandika jibu kama namba iliyochanganywa kwa sababu tatizo la awali lilitumia namba iliyochanganywa. Angalia: Je! Jumla ya tatu-nane na\(3 \dfrac{1}{8}\) sawa na tatu na nusu?

    Ongeza. \(3 \dfrac{4}{8} \stackrel{?}{=} 3 \dfrac{1}{2} \)
    Kurahisisha. \(3 \dfrac{1}{2} = 3 \dfrac{1}{2} \)

    Ufumbuzi hundi.

    Zoezi\(\PageIndex{27}\)

    Tafsiri na kutatua: Jumla ya tano na nane na\(x\) ni moja ya nne.

    Jibu

    \(\dfrac{5}{8}+x = \dfrac{1}{4}\);\(x = -\dfrac{3}{8}\)

    Zoezi\(\PageIndex{28}\)

    Tafsiri na kutatua: Tofauti ya moja na tatu-nne na\(x\) ni tano na sita.

    Jibu

    \(1\dfrac{3}{4} - x = \dfrac{5}{6}\);\(x = \dfrac{11}{12}\)

    Mazoezi hufanya kamili

    Kuamua Kama Fraction ni Suluhisho la Equation

    Katika mazoezi yafuatayo, onyesha kama kila nambari ni suluhisho la equation iliyotolewa.

    1. x -\(\dfrac{2}{5}\) =\(\dfrac{1}{10}\):
      1. x = 1
      2. x =\(\dfrac{1}{2}\)
      3. x =\(− \dfrac{1}{2}\)
    2. y -\(\dfrac{1}{2}\) =\(\dfrac{5}{12}\):
      1. y = 1
      2. y =\(\dfrac{3}{4}\)
      3. y =\(- \dfrac{3}{4}\)
    3. h +\(\dfrac{3}{4}\) =\(\dfrac{2}{5}\):
      1. h = 1
      2. h =\(\dfrac{7}{20}\)
      3. h =\(- \dfrac{7}{20}\)
    4. k +\(\dfrac{2}{5}\) =\(\dfrac{5}{6}\):
      1. k = 1
      2. k =\(\dfrac{13}{30}\)
      3. k =\(- \dfrac{13}{30}\)

    Kutatua Equations na Fractions kwa kutumia Kuongeza, Kutoa, na Idara ya Mali ya Usawa

    Katika mazoezi yafuatayo, tatua.

    1. y +\(\dfrac{1}{3}\) =\(\dfrac{4}{3}\)
    2. m +\(\dfrac{3}{8}\) =\(\dfrac{7}{8}\)
    3. f +\(\dfrac{9}{10}\) =\(\dfrac{2}{5}\)
    4. h +\(\dfrac{5}{6}\) =\(\dfrac{1}{6}\)
    5. a -\(\dfrac{5}{8}\) =\(- \dfrac{7}{8}\)
    6. c -\(\dfrac{1}{4}\) =\(- \dfrac{5}{4}\)
    7. x -\(\left(- \dfrac{3}{20} \right)\) =\(- \dfrac{11}{20}\)
    8. z -\(\left(- \dfrac{5}{12} \right)\) =\(- \dfrac{7}{12}\)
    9. n -\(\dfrac{1}{6}\) =\(\dfrac{3}{4}\)
    10. p -\(\dfrac{3}{10}\) =\(\dfrac{5}{8}\)
    11. s +\(\left(- \dfrac{1}{2} \right)\) =\(- \dfrac{8}{9}\)
    12. k +\(\left(- \dfrac{1}{3} \right)\) =\(- \dfrac{4}{5}\)
    13. 5j = 17
    14. 7k = 18
    15. -4w = 26
    16. -9v = 33

    Kutatua Equations na FRACTIONS Kutumia Mali ya Kuzidisha ya Usawa

    Katika mazoezi yafuatayo, tatua.

    1. \(\dfrac{f}{4}\)= -20
    2. \(\dfrac{b}{3}\)= -9
    3. \(\dfrac{y}{7}\)= -21
    4. \(\dfrac{x}{8}\)= -32
    5. \(\dfrac{p}{-5}\)= -40
    6. \(\dfrac{q}{-4}\)= -40
    7. \(\dfrac{r}{-12}\)= -6
    8. \(\dfrac{s}{-15}\)= 1-3
    9. -x = 23
    10. -y = 42
    11. -h =\(− \dfrac{5}{12}\)
    12. -k =\(− \dfrac{17}{20}\)
    13. \(\dfrac{4}{5}\)n = 20
    14. \(\dfrac{3}{10}\)p = 30
    15. \(\dfrac{3}{8}\)q = -48
    16. \(\dfrac{5}{2}\)m = -40
    17. \(- \dfrac{2}{9}\)a = 16
    18. \(- \dfrac{3}{7}\)b = 9
    19. \(- \dfrac{6}{11}\)u = -24
    20. \(- \dfrac{5}{12}\)v = -15

    Mazoezi ya mchanganyiko

    Katika mazoezi yafuatayo, tatua.

    1. 3x = 0
    2. 8y = 0
    3. 4f =\(\dfrac{4}{5}\)
    4. 7g =\(\dfrac{7}{9}\)
    5. p +\(\dfrac{2}{3}\) =\(\dfrac{1}{12}\)
    6. q +\(\dfrac{5}{6}\) =\(\dfrac{1}{12}\)
    7. \(\dfrac{7}{8}\)m =\(\dfrac{1}{10}\)
    8. \(\dfrac{1}{4}\)n =\(\dfrac{7}{10}\)
    9. \(- \dfrac{2}{5}\)= x +\(\dfrac{3}{4}\)
    10. \(- \dfrac{2}{3}\)= y +\(\dfrac{3}{8}\)
    11. \(\dfrac{11}{20}\)= -f
    12. \(\dfrac{8}{15}\)= -d

    Tafsiri sentensi kwa equations na Kutatua

    Katika mazoezi yafuatayo, tafsiri kwa equation ya algebraic na kutatua.

    1. Na kugawanywa na nane ni -16.
    2. n kugawanywa na sita ni -24.
    3. m imegawanywa na -9 ni -7.
    4. m imegawanywa na -7 ni -8.
    5. Quotient ya f na -3 ni 18-18.
    6. Quotient ya f na -4 ni -20.
    7. Kiwango cha g na kumi na mbili ni 8.
    8. Kiwango cha g na tisa ni 14.
    9. Tatu ya nne ya q ni 12.
    10. Mbili ya tano ya q ni 20.
    11. Sehemu ya kumi ya p ni -63.
    12. Nne ya tisa ya p ni -28.
    13. m kugawanywa na 4 sawa na hasi 6.
    14. Quotient ya h na 2 ni 43.
    15. Tatu ya nne ya z ni sawa na 15.
    16. quotient ya na\(\dfrac{2}{3}\) ni\(\dfrac{3}{4}\).
    17. Jumla ya tano na sita na x ni\(\dfrac{1}{2}\).
    18. Jumla ya tatu na nne na x ni\(\dfrac{1}{8}\).
    19. Tofauti ya y na moja ya nne ni\(- \dfrac{1}{8}\).
    20. Tofauti ya y na theluthi moja ni\(- \dfrac{1}{6}\).

    kila siku Math

    1. Ununuzi Teresa alinunua jozi ya viatu kuuzwa kwa $48. Bei ya kuuza ilikuwa\(\dfrac{2}{3}\) ya bei ya kawaida. Pata bei ya kawaida ya viatu kwa kutatua equation\(\dfrac{2}{3}\) p = 48
    2. Playhouse meza katika playhouse ya mtoto ni\(\dfrac{3}{5}\) ya meza ya watu wazima-size. Jedwali la kucheza ni urefu wa inchi 18. Pata urefu wa meza ya ukubwa wa watu wazima kwa kutatua equation\(\dfrac{3}{5}\) h = 18.

    Mazoezi ya kuandika

    1. Mfano 4.100 inaelezea mbinu tatu za kutatua equation -y = 15. Ni njia ipi unayopendelea? Kwa nini?
    2. Richard anadhani ufumbuzi wa equation\(\dfrac{3}{4}\) x = 24 ni 16. Eleza kwa nini Richard ni makosa.

    Self Check

    (a) Baada ya kukamilisha mazoezi, tumia orodha hii ili kutathmini ujuzi wako wa malengo ya sehemu hii.

    (b) Kwa ujumla, baada ya kuangalia orodha, unafikiri wewe ni vizuri tayari kwa ajili ya Sura ya? Kwa nini au kwa nini?

    Wachangiaji na Majina